Division

[cinelli]

Each asterisk represents a digit. What are the components of this long division?

.          *7***
       ---------
    ***)********
        ****
        ----
          ***
          ***
          ---
          ****
           ***
          ----
            ****
            ****
            ----
               0

Cinelli

[richfool]

Each asterisk represents a digit. What are the components of this long division?

.          *7***
       ---------
    ***)********
        ****
        ----
          ***
          ***
          ---
          ****
           ***
          ----
            ****
            ****
            ----
               0

Cinelli

Answer: ********

[Gengulphus]

Each asterisk represents a digit. What are the components of this long division?

.          *7***
       ---------
    ***)********
        ****
        ----
          ***
          ***
          ---
          ****
           ***
          ----
            ****
            ****
            ----
               0

Spoiler...

Using lower-case letters to denote digits (necessarily in the range 0-9) and upper-case letters to denote non-negative integers more generally, let the divisor be D and the quotient be a7bcd. From the long division, we can read off a number of facts, which include:

i) D is a 3-digit integer, i.e. 100 <= D <= 999;
ii) D*a is a 4-digit integer, i.e. 1000 <= D*a <= 9999;
iii) D*7 is a 3-digit integer, i.e. 100 <= D*7 <= 999, or 15 <= D <= 142;
iv) D*b is a 3-digit integer, i.e. 100 <= D*b <= 999;
v) D*c = 0, from which it follows that c = 0, since D is nonzero by i) above;
vi) D*d is a 4-digit integer, i.e. 1000 <= D*d <= 9999;
vii) D*a7bcd is an 8-digit integer, or 10000000 <= D*a7bcd <= 99999999;
viii) D*7bcd is a 6-digit integer, or 100000 <= D*7bcd <= 999999;
ix) D*bcd is a 6-digit integer, or 100000 <= D*bcd <= 999999;

From ii) and a <= 9, it follows that D >= RoundUp(1000/9) = 112. Combining that with iii), we know that 112 <= D <= 142. Using ii), iv) and vi), we can distinguish two cases within that: if 112 <= D <= 124, ii) implies that a=9, iv) implies that b <= 8 and vi) implies that d=9, while if 125 <= D <= 142, ii) implies that a is 8 or 9, iv) implies that b <= 7, and vi) implies that d is 8 or 9.

From v) and ix), we have 100000 <= D*b0d, from which it follows that 100000 - D*d <= D*b00 and dividing through by D, that RoundUp(100000/D) - d <= b00. If 112 <= D <= 123, we know that d=9 and so this implies that b00 >= RoundUp(100000/123) - 9 = 814 - 9 = 805, which is incompatible with the above deduction that b <= 8 for D in this range. Similarly, if 125 <= D <= 141, we know that d <= 9 and so this implies that b00 >= RoundUp(100000/141) - 9 = 710- 9 = 701, which is incompatible with b <= 7 in this range. So we're left with the possibilities that D=124 or D=142.

If D=142, we must have a = 8 or 9, d = 8 or 9 and b00 >= RoundUp(100000/142) - d = 705 - d = 696 or 697, which combines with b <= 7 to imply b=7. But this implies that D*7bcd >= 142 * 7708 = 1094536, which is incompatible with viii), and so does not lead to a solution.

We left with the possibility that D=124, which by the above implies that a=9, d=9 and b00 >= RoundUp(100000/124) - 9 = 807 - 9 = 798. The last combines with b <= 8 to imply b=8, so the quotient is 97809, the divisor is 124 and since the division comes out exactly, the dividend is 124 * 97809 = 12128316. All that remains is to check that this is indeed a solution:

.          97809
       ---------
    124)12128316
        1116
        ----
          968
          868
          ---
          1003
           992
          ----
            1116
            1116
            ----
               0

Gengulphus

[cinelli]

Full marks to Gengulphus for a flawless explanation.

Cinelli